Geometry of measures in real dimensions via H\"older parameterizations

TL;DR

This paper explores how non-integer Hausdorff densities influence the geometric structure of measures in Euclidean space, revealing conditions under which measures are supported on H"older or bi-Lipschitz curves, extending classical results.

Contribution

It extends geometric measure theory to non-integer dimensions by establishing parameterization theorems and measure classifications based on Hausdorff densities for real s between 0 and n.

Findings

Measures with positive lower and finite upper densities are supported on countably many bi-Lipschitz curves for 0<s<1.

Conditions on densities determine whether measures are supported on or singular to (1/s)-H"older curves for 1≤s<n.

Introduces parameterization theorems for sets with small Assouad dimension.

Abstract

We investigate the influence that $s$-dimensional lower and upper Hausdorff densities have on the geometry of a Radon measure in $\mathbb{R}^n$ when $s$ is a real number between $0$ and $n$. This topic in geometric measure theory has been extensively studied when $s$ is an integer. In this paper, we focus on the non-integer case, building upon a series of papers on $s$-sets by Mart\'in and Mattila from 1988 to 2000. When $0<s<1$, we prove that measures with almost everywhere positive lower density and finite upper density are carried by countably many bi-Lipschitz curves. When $1\leq s<n$, we identify conditions on the lower density that ensure the measure is either carried by or singular to $(1/s)$-H\"older curves. The latter results extend part of the recent work of Badger and Schul, which examined the case $s=1$ (Lipschitz curves) in depth. Of further interest, we introduce H\"older and bi-Lipschitz parameterization theorems for Euclidean sets with "small" Assouad dimension.

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